Exploring Excel spreadsheets in the teaching and learning of certain concepts of Statistical physics and Thermodynamics

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1 Exploring Excel spreadsheets in the teaching and learning of certain concepts of Statistical physics and Thermodynamics Ionel Grigore 1,2, Cristina Miron 1, Emil-Stefan Barna 1 (1) Faculty of Physics, University of Bucharest, Romania (2) Nichita Stănescu National College, Ploieşti, Romania grigore_1965[at]yahoo.com, cmiron_2001[at]yahoo.com, barna_emil[at]yahoo.com Abstract This paper demonstrates the way in which Excel spreadsheets can be explored in the teaching and learning of certain concepts of Statistical physics and Thermodynamics. It describes two didactic tools built with the help of spreadsheets in order for students to understand the model of the ideal gas more easily. The first instrument assists in the analysis of the Maxwell-Boltzman distribution according to the model of the speeds for an ideal gas, emphasizing certain particular aspects. The second instrument assists in the analysis of the isothermal process of an ideal gas and demonstrates the calculus relation for the work using the graphic interpretation of these measures within the pressure-volume diagram. Through the combined integration of the two tools in Physics lessons, students can grasp concepts such as distribution according to speeds, isothermal transformation and work in Thermodynamics. Also, by numerically calculating the work with the help of the facilities offered by the spreadsheet, students take an important step forward towards the understanding of a key concept of mathematical analysis, i.e. the definite integral. Keywords: Maxwell-Boltzmann distribution, isothermal transformation, work in thermodynamics, Physics Education. 1 Introduction The teaching and learning of Thermodynamics and Statistical physics constitute a research subject in Physics education. The literature presents articles analyzing the difficulties met by students in the study of Thermodynamics and Statistical physics and suggesting various solutions for an efficient approach of this Physics domain. The investigation of students understanding of the first principle of thermodynamics has shown that they do not frequently manage to differentiate between the concepts of heat, temperature, work and internal energy. The authors suggest that the incorrect interpretation of simple microscopic models can affect the understanding capacity of the macroscopic phenomena (Loverude et al, 2002). In order to guide students in solving thermodynamics problems, there have been designed applications in the C # language for the study of simple transformations of the ideal gas and for the analysis of basic thermodynamic cycles. With the help of these tools, students can gain a better understanding of the principles underlying thermodynamics and the way in which these principles can be used in solving problems (Liu, 2011). In the solving of thermodynamics problems spreadsheets have been promoted as tools that can be developed by students instead of other specialized programs (Sandler, 1997). The utilization of Excel formulas together with the programming means Visual Basic for Application (VBA) has generated models of thermodynamic systems and has shown the effects of the change in the input parameters on the final results (Caretto et al, 2005). Also, it has been demonstrated how the set of Solver functions from the Excel spreadsheet can be used to calculate the thermal balance for different substances known by minimizing the Gibbs energy. The results have been compared with those obtained by using a program written in the Fortran language finding a satisfying

2 400 University of Bucharest and West University of Timisoara concordance (Lwin, 2000). The graphic facilities of spreadsheets have rendered the construction of the phase diagrams for the binary systems Al-Zn and Li-Mg (Tomasini, 2014) and demonstrated how various aspects related to the Fermi-Dirac distribution can be clarified (Sharma and Ahluwalia, 2012). In the context of the classroom utilization of spreadsheets, the paper describes two Excel didactic tools designed for the teaching and learning of certain concepts specific to Statistical physics and Thermodynamics. The former allows the graphic visualization of the Maxwell-Boltzmann distribution according to the module of the speed for an ideal gas. We have graphically highlighted the typical speeds of the gas such as the most probable speed, the mean speed and the root mean square speed, together with the corresponding values of the distribution. The distribution according to speeds can be analyzed comparatively for different gases at different temperatures. In particular, there is the possibility to visualize the overlapping of the distribution curves according to the speeds of the same gas for three different temperatures or for three different gases at the same temperature. The latter allows the analysis of the isothermal process for an ideal gas. We have represented the isothermal curve in pressure-volume coordinates highlighting the initial and final states of the process and calculating the work changed by the gas with the external environment. Moreover, this tool helps demonstrate the calculation relation of the work in the isothermal process using the geometrical interpretation of this measure in the pressure-volume diagram. Thus, we have a comparison between the value of the work calculated analytically with the value of the work calculated numerically using the trapezoidal rule in the evaluation of a definite integral. 2 Organization of spreadsheets The structure of the tools presented in this paper is similar to that of other tools described by the authors which explore the facilities of Excel spreadsheets in the process of teaching and learning of Physics (Grigore et al, 2014; Grigore et al 2015). The main spreadsheet of each tool comprises the sections Data input and Results, plus the area of the associated graph. For each measure of the two sections we state the unit of measurement. We shall further describe how the mode of employment and the characteristics of each tool. Figure 1 renders the main spreadsheet of the tool for the analysis of the Maxwell-Boltzmann distribution according to speeds. Figure 1. The main spreadsheet of the tool for the analysis of the Maxwell-Boltzmann distribution according to speeds

3 The 10 th International Conference on Virtual Learning ICVL The measures introduced in the Data input section are: the universal gas constant R in cell B5, the molar mass µ in cell B6 and the gas temperature T in cell B7. In the Results section we calculate the most probable speed v p in cell B10, the mean speed v m in cell B11, the root mean square speed v T in cell B12. In order to perform the calculations in Excel we have used the following cell names: in the main spreadsheet Constant_R for cell B5, Mass_Mol for cell B6, Temperature for cell B7 and in the sheet for intermediary calculations, Constant_K for the cell in which we calculate Boltzmann constant k, and Mass for the cell in which we calculate the mass of the gas molecule, m 0. To perform the calculations leading to the analytic and graphic results from the main spreadsheet we have used the relations known in the literature (Serway and Jewett, 2013). For example, the distribution function according to the module of the speed is given by: 2 3 / 2 m0v m0 2 2 f( v) = 4πv e [1] kt 2πkT The graph from figure 1 renders in a red line the curve of the distribution function according to the module of the speed. It also renders the typical speeds of the gas with the values corresponding to the distribution in the dotted line segments colored differently. Thus, the most probable speed is highlighted in blue, the mean speed in pink and the root mean square speed in brown. The source table of this graph was drawn through a procedure used by the authors in other papers as well (Grigore et al, 2015). Thus, in column A of the table we have generated an increasing series with a unit step with which we further generated in column B the values of the speed. To generate the values of the speeds we have used a speed quantum equal to the 100 th part of the most probable speed v p and the value interval of the speed was fixed between 0 and 4v p. With the start values in the 4 th row of the spreadsheet, we have transcribed relation [1] in Excel in cell C4 as follows: =(Mass/(2*PI()*Constant_K*Temperature))^(3/2)*4*PI()*B4^2*EXP(-(Mass*B4^2)/ (2*Constant_K*Temperature)). Propagating the Excel formula along column C, there have resulted the values of the distribution according to the module of the speed. The source table of the graph in figure 1 also contains supplementary rows and columns to graphically highlight the velocities v p, v m, v T. Figure 2. Secondary spreadsheet for the comparative analysis of the distribution according to speeds. Comparison between the distributions of speeds of the same gas at three different temperatures

4 402 University of Bucharest and West University of Timisoara Figure 2 presents the spreadsheet with the comparative analysis for the distribution according to speeds when we have the same gas at different temperatures. Considering oxygen as an example of gas, O 2, we have introduced in cells A7, B7 and C7 the molar mass µ 1 =µ 2 =µ 3 =µ=32 g. In cells A10, B10 and C10 we have introduced three different temperatures, namely, T 1 =100 K, T 2 =300 K, T 3 =800 K. In the graph alongside the input data it can be observed how the distribution curve modifies according to the speeds with the temperature. As the temperature rises, the maximum value of the function f(v) decreases and the distribution curve moves towards greater values of the speed. Figure 3. Secondary spreadsheet for the comparative analysis of the distribution according to speeds. Comparison between the distributions according to speeds for three different gases at the same temperature Figure 3 renders the spreadsheet with the comparative analysis for the distribution according to speeds when we have different gases at the same temperature. Considering as examples of gases hydrogen, H 2, helium, He, and oxygen, O 2, we have introduced in cells A7, B7 and C7 the molar masses µ 1 =2 g, µ 2 =4 g, µ 3 =32 g. In cells A10, B10 and C10 we have introduced the same temperature T 1 =T 2 =T 3 =T=300 K. In the graph alongside the input data it can be observed how the distribution curve modifies according to the speeds together with the molar mass of the gas. It is checked that with the growth of the molar mass, there is a growth in the maximum value of the function f(v) and the distribution curve moves towards smaller and smaller values of the speed. The source tables of the graphs from figures 2 and 3 have been drawn in an analogy with the source table of the graph in figure 1. To generate the values of the speed in this case, we have used a speed quantum equal to v*/100, where v* represents the maximum of the most probable speeds of the three gases. The interval of values is between 0 and 3v*. With the help of relation [1] we have calculated the distribution function f(v) for the three gases, adapted to the new input data from the secondary sheet. Figure 4 renders the main spreadsheet of the tool for the analysis of the isothermal transformation of the ideal gas. The organization of the main spreadsheet of this tool is similar to that of the tool previously presented for the analysis of the Maxwell-Boltzmann distribution according to speeds for an ideal gas.

5 The 10 th International Conference on Virtual Learning ICVL Figure 4. The main spreadsheet of the tool for the analysis of the isothermal transformation of the ideal gas In the section Data input we introduce the following measures: the universal gas constant R in cell B4, the molar mass µ in cell B5, the mass of the gas m in cell B6, the temperature of the isothermal process T in cell B7, the volume of the initial state V 1 in cell B9, the volume of the final state V 2 in cell B10. In the section Results we calculate the difference of volume throughout the process V in cell B13, the pressure of the gas in initial state p 1 in cell B14, the pressure of the gas in final state p 2 in cell B15, the work done L in cell B16. To calculate the pressure in the initial and final states we have used the equation of state of the ideal gas and for the analytic calculation of the work the following relation (Serway and Jewett, 2013): [2] V 2 m V L = p( V ) dv = RT ln µ V V The domain A18:B21 from the Results section is reserved for the numeric calculation of the work using the geometric interpretation of this measure in p-v coordinates. In cell B19 we introduce the number of divisions n, in which the interval [V 1, V 2 ] is divided, in cell B20 we calculate the elementary interval δv=(v 2 -V 1 )/n, while in cell B21 we calculate the work L*, as the area below the isothermal curve corresponding to the interval [V 1, V 2 ]. The value of the work calculated numerically in cell B21 is compared to the value of the work calculated analytically according to relation [2] in cell B16. The associated graph from the main spreadsheet renders the isothermal curve of the process in the red line and highlights the pressure-volume pairs of values corresponding to the initial and final states through the segments of dotted line colored in blue. For the numeric calculation of the work we have drawn in a secondary sheet a table similar to the source table of the graph from figure 4. In column A of the table we have generated an

6 404 University of Bucharest and West University of Timisoara increasing series with a unit step from 0 to n=1000. In column B we have generated the values of the volume from the start value V 1 with the help of the series from column A, the volume quantum being determined by n, the number of divisions in which the interval [V 1, V 2 ] is divided. In column C we have calculated the pressure corresponding to each value of the volume from column B utilizing the equation of state. To perform the calculations we have introduced the following names of cells and domains: Pressure_1 for cell B14, Pressure_2 for cell B15, Delta_V for cell B20 in the main spreadsheet and Domain_P for the domain in which we calculate the values of the pressure in column C of the table from the secondary sheet. Applying the trapezoidal rule for the numeric evaluation of a definite integral, we have written the following Excel formulas in cell B21 of the main sheet: =(SUM(Domain_P)-(1/2)*(Pressure_1+Pressure_2))*Delta_V. It can be observed that the result in cell B21 coincides with the result from cell B16, L*=L. By modifying the number n of divisions of the interval [V 1, V 2 ] and correspondingly redrawing the source table, the effect on the result displayed in cell B21 can be observed. It is thus checked that with the increase of n, the value of the work in cell B21 tends towards the value of the work in cell B16. 4 Conclusions Students participation in the creation of the didactic tools described in this paper can be beneficial for the understanding of the model of the ideal gas. The classroom utilization of the respective tools represents a conclusive example of how to emphasize the facilities offered by the Excel program. The first tool helps students clarify aspects connected to the Maxwell-Boltzmann distribution according to speeds of the ideal gas and rapidly observe the effect on the distribution function when modifying the input data. Thus, the change in the distribution can be studied according to speeds when the temperature and/or the molar mass of the gas changes. The second tool facilitates students understanding of the isothermal transformation of the ideal gas and clarifies certain aspects connected to the concept of work in thermodynamics. Thus, this tool offers students an alternative demonstration for the calculus relation of the work in the isothermal transformation of the ideal gas. This demonstration based on the geometrical interpretation of the work within the pressure-volume diagram employs the data manipulation capacity in the spreadsheet. The numeric calculation of the work represents an important step towards the assimilation of a key concept of mathematical analysis, namely the concept of definite integral. Students can better observe the manner in which the knowledge transfer between Mathematics and Physics takes place. References Serway, R.A., Jewett, J.W. Jr. (2013): Physics for Scientists and Engineers with Modern Physics, 9 th Edition, Thomson Brooks/Cole, ISBN-13: Grigore, I., Miron, C., Barna, E.S. (2015) Using Microsoft Excel in Teaching and Learning Relativistic Kinematics, Romanian Reports in Physics, 67, 2, Liu, Y. (2011) Development of Instructional Courseware in Thermodynamics Education, Computer Applications in Engineering Education, 19, Loverude, M.E., Kautz, C.H., Heron, Paula R.L. (2002) Student understanding of the first law of thermodynamics: Relating work to the adiabatic compression of an ideal gas, American Journal of Physics, 70, 2, Lwin, Y. (2000) Chemical Equilibrium by Gibbs Energy Minimization on Spreadsheets. Int. J. Engng Ed., 16, 4,

7 The 10 th International Conference on Virtual Learning ICVL Sandler, S.I. (1997) Spreadsheets for Thermodynamics Instruction: Another Point of View, Chemical Engineering Education (CEE), 31, 1, Sharma, S., Ahluwalia, P.K. (2012) Diagnosing alternative conceptions of Fermi energy among undergraduate students, European Journal of Physics, 33, Tomasini, P. (2014) Deconstructing Phase Diagram Calculations. Journal Of Chemical Education, 91, Caretto, L., McDaniel, D., Mincer, T. (2005): Spreadsheet Calculations of Thermodynamic Properties, In Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition Copyright, Paper Grigore, I., Miron, C., Barna, E.S. (2014): Using Excel spreadsheets to process data in Physics didactic experiments, In Proceedings of The 9 th International Conference On Virtual Learning, Bucharest, October 24-25, 2014,